Abstract
The authors establish several estimates showing that the distance in W1,p, 1 < p < ∞, between two immersions from a domain of Rn into Rn+1 is bounded by the distance in Lp between two matrix fields defined in terms of the first two fundamental forms associated with each immersion. These estimates generalize previous estimates obtained by the authors and P. G. Ciarlet and weaken the assumptions on the fundamental forms at the expense of replacing them by two different matrix fields.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R., Marsden, J. E. and Ratiu, T., Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York, 1988.
Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.
Ciarlet, P. G., An Introduction to Differential Geometry, with Applications to Elasticity, Springer-Verlag, Heidelberg, 2005.
Ciarlet, P. G., Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013.
Ciarlet, P. G., Gratie, L. and Mardare, C., A nonlinear Korn inequality on a surface, J. Math. Pures Appl., 85, 2006, 2–16.
Ciarlet, P. G., Malin M. and Mardare, C., New nonlinear estimates for surfaces in terms of their fundamental forms, C. R. Acad. Sci. Paris, Ser. I, 355, 2017, 226–231.
Ciarlet, P. G. and Mardare, C., Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl., 83, 2004, 811–843.
Ciarlet, P. G. and Mardare, C., Nonlinear Korn inequalities, J. Math. Pures Appl., 104, 2015, 1119–1134.
Conti, S., Low-Energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns, Habilitationsschrift, Universität Leipzig, 2004.
Friesecke, G., James, R. D. and Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math., 55, 2002, 1461–1506.
Klingenberg, W., A Course in Differential Geometry, Springer-Verlag, Berlin, 1978.
Kühnel, W., Differential Geometry: Curves-Surfaces-Manifolds, American Mathematical Society, Providence, 2002.
Malin, M. and Mardare, C., Nonlinear estimates for hypersurfaces in terms of their fundamental forms, C. R. Acad. Sci. Paris, Ser. I, 355, 2017, 1196–1200.
Nečas, J., Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967.
Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36, 1934, 63–89.
Acknowledgements
The authors would like to thank Professor Philippe G. Ciarlet for stimulating discussions that initiated this research and for his hospitality at the City University of Hong Kong. M. Malin is also very grateful to the City University of Hong Kong for its support during the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Research Grants Council of the Hong Kong Special Administration Region, China (No. 9042388, CityU 11305716).
Rights and permissions
About this article
Cite this article
Malin, M., Mardare, C. Nonlinear Korn Inequalities on a Hypersurface. Chin. Ann. Math. Ser. B 39, 513–534 (2018). https://doi.org/10.1007/s11401-018-0080-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-018-0080-x